Optimal decay rate of the bipolar Euler-Poisson system with damping in R³

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By rewriting a bipolar Euler-Poisson equations with damping into an Euler equation with damping coupled with an Euler-Poisson equation with damping, and using a new spectral analysis, we obtain the optimal decay results of the solutions in L²-norm, which improve theose in Li3, Wu3. More precisely, the velocities u₁,u₂ decay at the L²-rate (1+t)-54, which is faster than the normal L²-rate (1+t)-34 for the Heat equation and the Navier-Stokes equations. In addition, the disparity of two densities ρ₁-ρ₂ and the disparity of two velocities u₁-u₂ decay at the L²-rate (1+t)⁻².